Great Circle

Author

Eric W. Weisstein
November 29, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/GreatCircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GreatCircle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Radius Vector

r[λ_,δ_]:=a{Cos[λ]Cos[δ],Sin[λ]Cos[δ],Sin[δ]}

Simplify[Sqrt[#.#&@r[λ,δ]],a>0]

Angular distance

ArcCos[Simplify[r[(Subscript[λ, 1]),(Subscript[δ, 1])].r[(Subscript[λ, 2]),(Subscript[δ, 2])]/a^2]]

Geodesic

r[u,Pi/2-v]

Ds[f_,{u_,v_}]:=Plus@@({D[#,u]^2,D[#,u]D[#,v],D[#,v]^2}&/@f)

{P,Q,R}=Ds[r[u,Pi/2-v],{u,v}]//Simplify

i=FullSimplify[c1 Sqrt[R/(P^2-c1^2 P)],c1>0&&a>0&&0<v<π]

v4.x

Set b=a/(Subscript[c, 1])to avoid Integrate giving a needlessly complicated answer

FullSimplify[Integrate[(1/(Sin[v](Sqrt[b^2  ((Sin[v])^2)- 1]))),v]]

FullSimplify[-ArcTan[(((Sqrt[2]) Cos[v])/(Sqrt[Expand[-2+(b^2)-(b^2) (1-2((Sin[v])^2))]]))]]

Which can be rewritten as

FullSimplify[ArcTan[((Cos[v])/(Sqrt[-1+(b^2) ((Sin[v])^2)]))]-ArcSin[((Cot[v])/(Sqrt[b^2-1]))],b>0&&v>0]

FunctionExpand[-ArcSin[((Cot[v])/(Sqrt[-1+(b^2)]))]+ArcTan[((Cos[v])/(Sqrt[-1+(b^2) ((Sin[v])^2)]))]]

Plot[Evaluate[ArcTan[((Cos[v])/(Sqrt[-1+(b^2) ((Sin[v])^2)]))]-ArcSin[((Cot[v])/(Sqrt[b^2-1]))]]/.b->1.3,{v,0,Pi}]

v5.2

Integrate[i,v,Assumptions->0<a<c1]

FullSimplify[%,0<a<c1&&0<v<π]

FullSimplify[%/.Cos[2v]:>1-2Sin[v]^2,0<a<c1&&0<v<π]

FullSimplify[I/Sqrt[-c1^2+a^2*Sin[v]^2],0<a<c1&&0<v<Pi]

FullSimplify[%235/. 1/Sqrt[x_]:>I/ Sqrt[-x],0<a<c1&&0<v<Pi]

Minus sign problem here...

Intermediate Variable

Integrate[(1/(Sin[v](Sqrt[(b^2)((Sin[v])^2)-1]))),v,Assumptions->b>0&&0<v<π]

TrigExpand[%220]

Simplify[Simplify[TrigExpand//@%220]/.Cos[2v]:>1-2Sin[v]^2]